The function $g$ is continuous over $[a,b]$. I define a "trouble point" $x$ as follows: $\forall \delta>0$, the interval $[x-\delta,x+\delta]$ contains both points where $g$ is negative and points where $g$ is positive, as well as points $y \neq x$ for which $g(y)=0$. For example, $x=0$ is a "trouble point" for $g(x)=x\sin(\frac 1 x)$. $a$ and $b$ are considered "trouble points" if the same rule applies for $[a,a+\delta]$ and $[b-\delta,b]$, respectively.
My question is, if $g$ changes signs an infinite number of times on $[a,b]$, can there be an infinite number of "trouble points" on $[a,b]$, or must there be a finite number?
Hint: the set of trouble points can't be dense (do you see why?). However, if you take a discrete infinite subset of $[0, 1]$ say, $X = \{1, 1/2 , 1/3, 1/4, \ldots\}$, you can construct a continuous function $f$ that changes sign infinitely often near each $y \in X$ (modelled on the $x \sin(x)$ pattern adjusted and scaled to be $0$ for points $x$ that are not near $y$).